Let $(M,g)$ be a complete Riemannian, connected, compact manifold (with or without boundary). Let $f(r)$ be a decreasing function of $r =$ geodesic distance. If $\Omega \subset M$, then $$ \int_{\Omega} f(r)\, dV \leq \int_{\Omega^\star} f(r)\, dV \, ,$$ where $\Omega^\star$ is a geodesic ball with the same volume as $\Omega$.

Is that true? What if $\Omega$ is large? How can one define a geodesic ball, say if the radius is larger than $\mathrm{inj}(M,g)$?

Let $(M,g)$ be a complete Riemannian, connected, compact manifold (with or without boundary). Let $f(r)$ be a decreasing function of $r =$ geodesic distance. If $\Omega \subset M$, then $$ \int_{\Omega} f(r)\, dV \leq \int_{\Omega^\star} f(r)\, dV \, ,$$ where $\Omega^\star$ is a geodesic ball with the same volume as $\Omega$.

Is that true? What if $\Omega$ is large? How can one define a geodesic ball, say if the radius is larger than $\mathrm{inj}(M,g)$?

Addentum:

After reading the argument given below, there is still something that seems unclear to me. In order to use the co-area formula, one needs to have certain hypotheses, namely smoothness of functions (at least, the statement for manifold I know is for smooth functions : http://en.wikipedia.org/wiki/Smooth_coarea_formula). In particular, when claiming that $|∇r|=1$ in the argument below, it seems true only for $r<injrad(M,g)$. I believe (though I do not have a reference) that the distance function on a complete manifold is Lipschitz, thus differentiable almost everywhere by Rademacher's Theorem. But even with that, I am not sure this is enough to apply the coarea formula on a MANIFOLD.

Let $(M,g)$ be a complete connected compact manifold (with or without boundary). Let $f(r)$ be a decreasing function of $r =$ geodesic distance. If $\Omega \subset M$, then

$$ \int_{\Omega} f(r) dV \leq \int_{\Omega^\star} f(r) dV,$$

where $\Omega^\star$ is a geodesic ball of same volume than $\Omega$.

Is that true?

What if $\Omega$ is large, how can one define a geodesic ball (say if the radius is larger than $inj(M,g)$).

Let $(M,g)$ be a complete Riemannian, connected, compact manifold (with or without boundary). Let $f(r)$ be a decreasing function of $r =$ geodesic distance. If $\Omega \subset M$, then $$ \int_{\Omega} f(r)\, dV \leq \int_{\Omega^\star} f(r)\, dV \, ,$$ where $\Omega^\star$ is a geodesic ball with the same volume as $\Omega$.

Is that true? What if $\Omega$ is large? How can one define a geodesic ball, say if the radius is larger than $\mathrm{inj}(M,g)$?